Level set

A level set of a function $f$ is a set of the form $\{x:f(x)=c\}$.

A level set of a real-valued function on three-dimensional space. Where the function is continuously differentiable and its gradient is nonzero, the regular level set theorem makes this a smooth surface with normal vector parallel to the gradient. The equation alone can also have singular points, or fail to describe a two-dimensional surface at all.

For a fixed nonzero $a\in {\mathbb{R}}^n$, a nonzero level set $f(x)=h$ is the sphereThis follows by completing the square after multiplying by the positive denominator. Nonzero levels exist only for $|h|\le \Vert a\Vert /2$. The zero level is the hyperplane $a\cdot x=0$; the extreme levels shrink to points $x=\pm a/\Vert a\Vert$. The Cauchy-Schwarz inequality gives $|f(x)|\le \Vert a\Vert \Vert x\Vert /(1+\Vert x{\Vert }^2)\le \Vert a\Vert /2$, so these extrema are global. At them the Hessian matrix is respectively $-\Vert a\Vert I/2$ and $+\Vert a\Vert I/2$. In two dimensions the spheres are circles whose centres all lie on the axis spanned by $a$.

A level curve is the planar level set of a scalar function. Where the gradient does not vanish, the implicit function theorem makes it locally a smooth curve. At a critical point, branches may cross, narrow into a cusp or shrink to a point, so a plot should be interpreted with the local critical-point analysis.

A contour plot displays several level curves of a real-valued function of two variables, usually with labels indicating their values. It represents a surface's height on a flat page. Closely spaced level curves indicate a large change per unit distance only when the gaps between their labeled values are comparable.

A level set at a regular value is a smooth submanifold. By the regular level set theorem, its codimension is the dimension of the target. For determinant-one real symmetric matrices, $D(\det {)}_B(C)=\mathrm{tr}(B^{-1}C)$ is nonzero because $C=B$ gives $n$, so the constraint has codimension one.

A superlevel set of a real-valued function $f$ is a set of the form $\{x:f(x)\ge c\}$ or $\{x:f(x)>c\}$.